Sim
Start with 6 dots on a piece of paper. They do not have to be in a circle, but it does make the diagram neat. Two players Blue and Red take turns to join a pair of dots with a line of their colour. The first player to form a triangle using only their colour loses! In the picture at the top it is Red's turn to draw a red line. They should not join the pair of dots that would make a red triangle. Instead, they can join any other pair of dots that are not already joined. It would then be Blue's turn.
It is a great game to play with kids as a follow up to Noughts and Crosses. It is also a completely solved game, but that does not stop it from being fun and tricky. A draw is not possible because of the mathematical theory given below. Would you want to go first or second?
Where did the game come from . . . .
In mathematics, the word graph is used in two very different ways. The most common variety is taught in schools using x and y coordinates, such as y = x2,
and in statistics, such as a bar chart showing the results of the sum of a pair of dice thrown many times.
However, the term graph is also used for a network of dots joined by lines, such as a map of the London Underground, where the dots represent stations.
The exact position of the dots and whether the lines are straight or perhaps cross, does not matter. The important information is how they are connected.
In a complete graph with 6 dots, called K6, every dot is joined to every other dot, as shown here. The lines did not have to be straight and accidental intersections of lines are not counted as dots.
If you colour all the lines of K6 using 2 colours in any arbitrary manner, then it can be shown that there must be a set of 3 dots connected by the same colour. In the picture shown, it would mean there must be a triangle with edges of only one colour. Such graphs are not studied in school and the proof of this property has often been set as a stretching question for Oxbridge maths applicants. (See the note below.) It is also the source of the simple game called Sim.
Modern, fast computers can easily play through all possible games of Sim. It follows that it is know whether the first or second player can always win, with perfect play - but I am not telling you here! However, even with this knowledge, there is no simple strategy to win, so a game between humans, especially children, can be very challenging.
N.B. The Oxbridge maths task is often set as "In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances." The problem is made more accessible by drawing a graph with dots representing the six people and using a blue line to join each pair of acquaintances and a red line to join each pair of strangers.